Lattice Points in Polytopes

A lattice polygon
P : lattice polygon in R 2
(vertices ∈ Z 2, no self-intersections)
Lattice Points in Polytopes – p. 2
Boundary & interior lattice points
Lattice Points in Polytopes – p. 3
Pick’s theorem
Then
Pick’s theorem
Then
2 = 9.
Two tetrahedra
Pick’s theorem (seemingly) fails in higher dimensions. For example, let T1 and T2 be the tetrahedra with vertices
vert(T1) = {(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)} vert(T2) = {(0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1)}.
Lattice Points in Polytopes – p. 5
Failure of Pick’s theorem in dim 3
Then I(T1) = I(T2) = 0
Convex hull
conv(S) =
Convex hull
conv(S) =
Convex hull
conv(S) =
Polytope dilation
Let P be a convex polytope (convex hull of a finite set of points) in R
d. For n ≥ 1, let
nP = {nα : α ∈ P}.
Polytope dilation
Let P be a convex polytope (convex hull of a finite set of points) in R
d. For n ≥ 1, let
nP = {nα : α ∈ P}.
i(P, n)
the number of lattice points in nP.
Lattice Points in Polytopes – p. 9
i(P, n)
Similarly let
i(P, n) = #(nP ∩ Z d)
= #{α ∈ P : nα ∈ Z d},
the number of lattice points in the interior of nP.
Lattice Points in Polytopes – p. 10
An example
P 3P
Reeve’s theorem
lattice polytope: polytope with integer vertices
Theorem (Reeve, 1957). Let P be a three-dimensional lattice polytope. Then the volume V (P) is a certain (explicit) function of i(P, 1), i(P, 1), and i(P, 2).
Lattice Points in Polytopes – p. 12
Reeve’s theorem
lattice polytope: polytope with integer vertices
Theorem (Reeve, 1957). Let P be a three-dimensional lattice polytope. Then the volume V (P) is a certain (explicit) function of i(P, 1), i(P, 1), and i(P, 2).
Recall: i(P, 1) = number of interior lattice points.
Lattice Points in Polytopes – p. 12
The main result
P = lattice polytope in R N , dimP = d.
Then i(P, n) is a polynomial (the Ehrhart polynomial of P) in n of degree d.
Lattice Points in Polytopes – p. 13
Reciprocity and volume
(reciprocity).
Reciprocity and volume
(reciprocity).
where V (P) is the volume of P.
Lattice Points in Polytopes – p. 14
Eugène Ehrhart
1932: begins teaching career in lycées
1959: Prize of French Sciences Academy
1963: begins work on Ph.D. thesis
1966: obtains Ph.D. thesis from Univ. of Strasbourg
1971: retires from teaching career
January 17, 2000: dies
Photo of Ehrhart
Self-portrait
Generalized Pick’s theorem
Corollary. Let P ⊂ R d and dimP = d. Knowing
any d of i(P, n) or i(P, n) for n > 0 determines V (P).
Lattice Points in Polytopes – p. 18
Generalized Pick’s theorem
Corollary. Let P ⊂ R d and dimP = d. Knowing
any d of i(P, n) or i(P, n) for n > 0 determines V (P).
Proof. Together with i(P, 0) = 1, this data determines d + 1 values of the polynomial i(P, n) of degree d. This uniquely determines i(P, n) and hence its leading coefficient V (P).
Lattice Points in Polytopes – p. 18
An example: Reeve’s theorem
Example. When d = 3, V (P) is determined by
i(P, 1) = #(P ∩ Z 3)
i(P, 2) = #(2P ∩ Z 3)
i(P, 1) = #(P ∩ Z 3),
which gives Reeve’s theorem.
Lattice Points in Polytopes – p. 19
Birkhoff polytope
polytope of all M × M doubly-stochastic matrices A = (aij), i.e.,
aij ≥ 0
∑
Lattice Points in Polytopes – p. 20
(Weak) magic squares
Note. B = (bij) ∈ nBM ∩ Z M×M if and only if
bij ∈ N = {0, 1, 2, . . . } ∑
i
Example of a magic square





Example of a magic square





HM(n)
Lattice Points in Polytopes – p. 23
HM(n)
H1(n) = 1
HM(n)
H1(n) = 1
The caseM = 3
Values for smalln
Values for smalln
Values for smalln
Values for smalln
Values for smalln
Values for smalln
Anand-Dumir-Gupta conjecture
Theorem (Birkhoff-von Neumann). The vertices of BM consist of the M ! M × M permutation matrices. Hence BM is a lattice polytope.
Lattice Points in Polytopes – p. 26
Anand-Dumir-Gupta conjecture
Theorem (Birkhoff-von Neumann). The vertices of BM consist of the M ! M × M permutation matrices. Hence BM is a lattice polytope.
Corollary (Anand-Dumir-Gupta conjecture). HM(n) is a polynomial in n (of degree (M − 1)2).
Lattice Points in Polytopes – p. 26
H4(n)
+40950n + 11340) .
Reciprocity for magic squares
Reciprocity ⇒ ±HM(−n) =
#{M×M matrices B of positive integers, line sum n}. But every such B can be obtained from an M × M matrix A of nonnegative integers by adding 1 to each entry.
Lattice Points in Polytopes – p. 28
Reciprocity for magic squares
Reciprocity ⇒ ±HM(−n) =
#{M×M matrices B of positive integers, line sum n}. But every such B can be obtained from an M × M matrix A of nonnegative integers by adding 1 to each entry.
Corollary.
HM(−M − n) = (−1)M−1HM(n)
Lattice Points in Polytopes – p. 28
Two remarks
Lattice Points in Polytopes – p. 29
Zeros ofH9(n) in complex plane
Zeros of H_9(n)
Zeros ofH9(n) in complex plane
Zeros of H_9(n)
Coefficients ofi(P, n)
Coefficients of nd, nd−1, and 1 are “nice”, well-understood, and positive.
Lattice Points in Polytopes – p. 31
Coefficients ofi(P, n)
Coefficients of nd, nd−1, and 1 are “nice”, well-understood, and positive.
Let P denote the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 13). Then
i(P, n) = 13
The “bad” tetrahedron
The “bad” tetrahedron
z
x
y
Thus in general the coefficients of Ehrhart polynomials are not “nice.” There is a better basis (not given here).
Lattice Points in Polytopes – p. 32
Zonotopes
generated by v1, . . . , vk:
Lattice Points in Polytopes – p. 33
Zonotopes
generated by v1, . . . , vk:
Z(v1, . . . , vk) = {λ1v1 + · · · + λkvk : 0 ≤ λi ≤ 1} Example. v1 = (4, 0), v2 = (3, 1), v3 = (1, 2)
(4,0)
Lattice points in a zonotope
Theorem. Let
where vi ∈ Z d. Then the coefficient of nj in
i(Z, n) is given by ∑
X h(X), where X ranges over all linearly independent j-element subsets of {v1, . . . , vk}, and h(X) is the gcd of all j × j minors of the matrix whose rows are the elements of X.
Lattice Points in Polytopes – p. 34
An example
Example. v1 = (4, 0), v2 = (3, 1), v3 = (1, 2)
(4,0)
v1 = (4, 0), v2 = (3, 1), v3 = (1, 2)
i(Z, n) =
+gcd(1, 2))n + det(∅) = (4 + 8 + 5)n2 + (4 + 1 + 1)n + 1
= 17n2 + 6n + 1.
v1 = (4, 0), v2 = (3, 1), v3 = (1, 2)
i(Z, n) =
+gcd(1, 2))n + det(∅) = (4 + 8 + 5)n2 + (4 + 1 + 1)n + 1
= 17n2 + 6n + 1.
Corollaries
Corollary. If Z is an integer zonotope generated by integer vectors, then the coefficients of i(Z, n) are nonnegative integers.
Lattice Points in Polytopes – p. 37
Corollaries
Corollary. If Z is an integer zonotope generated by integer vectors, then the coefficients of i(Z, n) are nonnegative integers.
Neither property is true for general integer polytopes. There are numerous conjectures concerning special cases.
Lattice Points in Polytopes – p. 37
The permutohedron
Lattice Points in Polytopes – p. 38
The permutohedron
dim Πd = d − 1, since ∑
w(i) =
The permutohedron
dim Πd = d − 1, since ∑
w(i) =
Lattice Points in Polytopes – p. 38
Π3
321
312
213
123
132
Π3
321
312
213
123
132
Lattice Points in Polytopes – p. 39
Π4
i(Πd, n)
k=0 fk(d)xk, where
Lattice Points in Polytopes – p. 41
i(Πd, n)
k=0 fk(d)xk, where
fk(d) = #{forests with k edges on vertices 1, . . . , d} 1 2
3
Lattice Points in Polytopes – p. 41
Application to graph theory
Let G be a graph (with no loops or multiple edges) on the vertex set V (G) = {1, 2, . . . , n}. Let
di = degree (# incident edges) of vertex i.
Define the ordered degree sequence d(G) of G by
d(G) = (d1, . . . , dn).
Example ofd(G)
1 2
# of ordered degree sequences
Let f(n) be the number of distinct d(G), where V (G) = {1, 2, . . . , n}.
Lattice Points in Polytopes – p. 44
f(n) for n ≤ 4
Example. If n ≤ 3, all d(G) are distinct, so f(1) = 1, f(2) = 21 = 2, f(3) = 23 = 8. For n ≥ 4 we can have G 6= H but d(G) = d(H), e.g.,
3 4
The polytope of degree sequences
Let conv denote convex hull, and
Dn = conv{d(G) : V (G) = {1, . . . , n}} ⊂ R n,
the polytope of degree sequences (Perles, Koren).
Lattice Points in Polytopes – p. 46
The polytope of degree sequences
Let conv denote convex hull, and
Dn = conv{d(G) : V (G) = {1, . . . , n}} ⊂ R n,
the polytope of degree sequences (Perles, Koren).
Easy fact. Let ei be the ith unit coordinate vector in R
n. E.g., if n = 5 then e2 = (0, 1, 0, 0, 0). Then
Dn = Z(ei + ej : 1 ≤ i < j ≤ n).
Lattice Points in Polytopes – p. 46
The Erdos-Gallai theorem
Then α = d(G) for some G if and only if
α ∈ Dn
Lattice Points in Polytopes – p. 47
A generating function
A formula for F (x)
F (x) = 1
49-1
Two references
M. Beck and S. Robins, Computing the Continuous Discretely, Springer, 2010.
Lattice Points in Polytopes – p. 50
Two references
M. Beck and S. Robins, Computing the Continuous Discretely, Springer, 2010. ??, Enumerative Combinatorics, vol. 1, 2nd ed. (Sections 4.5–4.6), Cambridge Univ. Press, 2011.
Lattice Points in Polytopes – p. 50
Lattice Points in Polytopes – p. 51
ma {A lattice polygon}
ma {Pick's theorem}
ma {Pick's theorem}
ma {Two tetrahedra}
ma {Convex hull}
ma {Convex hull}
ma {Convex hull}
ma {Polytope dilation}
ma {Polytope dilation}
ma {An example}
ma {Reeve's theorem}
ma {Reeve's theorem}
ma {Birkhoff polytope}
ma {$m {H_M(n)}$}
ma {$m {H_M(n)}$}
ma {$m {H_M(n)}$}
ma {Values for small $n$}
ma {Values for small $n$}
ma {Values for small $n$}
ma {Values for small $n$}
ma {Values for small $n$}
ma {Values for small $n$}
ma {Anand-Dumir-Gupta conjecture}
ma {Anand-Dumir-Gupta conjecture}
ma {$m {H_4(n)}$}
ma {Two remarks}
ma {Coefficients of $m {i(cp ,n)}$}
ma {Coefficients of $m {i(cp ,n)}$}
ma {The ``bad'' tetrahedron}
ma {The ``bad'' tetrahedron}
ma {An example}
ma {Corollaries}
ma {Corollaries}
ma {$m {f(n)}$ for $m {nleq 4}$}
ma {The polytope of degree sequences}
ma {The polytope of degree sequences}
ma {The ErdH {o}s-Gallai theorem}
ma {A generating function}
ma {Two references}
ma {Two references}